Portfolio Theory

FIN 223 Lecture 1

Andrew Ainsworth

University of Wollongong

Lecture Outline

  • A quick refresher on return and risk measurement
  • Investor preferences and utility
    • How do we model investor’s choices between different risky portfolios?
  • Portfolio theory
    • What happens when investors combine different assets into portfolios?
    • What are diversification benefits?
  • Creating optimal portfolios
    • What portfolios do investors choose to maximize their utility?
  • Reading
    • Ch 5.3, 5.5, 5.6
    • Ch. 6.1-6.4

Let’s get started

Scenario return and risk

  • Expected returns \((E(R))\) are defined as: \[E(R) = \sum_{s=1}^K p_s \times R_s\]

    • \(s\) represents an economic scenario ranging from 1, 2, 3 … to \(K\)
      • The scenarios could be recession, boom or normal economic times
    • \(p_s\) is the probability of a certain scenario (\(s\)) occuring
    • \(R_s\) is the asset return if a certain scenario (\(s\)) occurs
  • Variance is defined as: \[\sigma^2=\sum_{s=1}^K p_s \times [R_s-E(R)]^2\]

  • Standard deviation \((\sigma)\) = \(\sqrt{\sigma^2}\)

Scenario return and risk

Scenario Probability Return
Excellent 25%   31.00%
Good 45%   14.00%
Poor 25%   −6.75%
Crash  5%  −52.00%

\[ \begin{align} E(R) &= \sum_{s=1}^4 p_s \times R_s \\ &= (0.25 \times 0.31) + (0.45 \times 0.14) + (0.25 \times -0.0675) + (0.05 \times -0.52) \\ &= 0.0976 \\ &=9.76\% \end{align} \]

Scenario return and risk

Scenario Probability Return
Excellent 25%   31.00%
Good 45%   14.00%
Poor 25%   −6.75%
Crash  5%  −52.00%

\[ \begin{align} \sigma^2 &= \sum_{s=1}^4 p_s \times [R_s-E(R)]^2 \\ &= 0.25 \times (0.31-0.0976)^2 + 0.45 \times (0.14-0.0976)^2 + \\ & \qquad 0.25 \times (-0.0675-0.0976)^2 + 0.05 \times (-0.52-0.0976)^2 \\ &= 0.038\\ \sigma &= \sqrt{0.038}=0.1949=19.49\%\\ \end{align} \]

Arithmetic and geometric averages

  • Arithmetic average
    • If we assume the probability of each state is equal then \(p_s=\frac{1}{K}\): \[E(R) = \sum_{s=1}^K p_s \times R_s=\frac{1}{K}\sum_{s=1}^K R_s\]
  • Geometric (time-weighted) average
    • Terminal value (TV) of the investment: \[TV_n = (1+r_1)(1+r_2)...(1+r_n)\]
    • Geometric average return \((g)\) over \(n\) time periods: \[g=TV_n^{1/n} -1\]

Estimating variance and standard deviation

  • Population variance is the measured as the expected value of squared deviations \[\sigma^2 = \frac{1}{n} \sum_{t=1}^n [R_t-\bar{R}]^2\]
  • We want to work with the sample variance and standard deviation so need to divide by \(n-1\) rather than \(n\) \[\sigma^2 = \frac{1}{n-1} \sum_{t=1}^n [R_t-\bar{R}]^2\] \[\sigma = \sqrt{\frac{1}{n-1} \sum_{t=1}^n [R_t-\bar{R}]^2}\]

The normal distribution

  • Investment analysis can “appear” easier if we assume asset returns follow a normal distribution
    • Symmetric asset returns \(\rightarrow\) Standard deviation is a good measure of risk
    • Symmetric asset returns \(\rightarrow\) Portfolio returns will be symmetric as well
    • Only mean and standard deviation needed to estimate future scenarios
    • Pairwise correlation coefficients summarize the dependence of returns across securities
  • You need to be aware that we are making an assumption when we focus only on means and standard deviation

The normal distribution

Investor decision making

Investor decision making

  • How does an investor choose the amount to invest in risky assets?
    • We can use a utility function to rank alternative outcomes
    • Investors choose among different assets or portfolios to maximise expected utility
    • Investors can have different levels of risk aversion
    • How much do they dislike risk?
    • The utility function will lead to different portfolio choices by investors
  • If there is no uncertainty then we just need to determine how much we want to consume now, and how much we want to consume later
    • Risk-free asset
  • If return is not certain across all possible states of the world
    • The range of possible future cash flows will impact on wealth
    • Risky assets

Portfolio risk and return

Portfolio with two risky assets

  • For a portfolio with two risky assets, the expected return and variance equations are: \[E(R_p) = w_1 E(R_1) + w_2 E(R_2)\]

  • Correlation coefficient between two assets:

  • Need to estimate two expected return, two variances and one covariance/correlation

How do investors choose portfolios?

Optimal combined portfolio

  • Choosing the optimal risky portfolio
    • Best combination of risky and safe assets to form portfolio
    • Optimal portfolio CAL tangent to efficient frontier
    • Slope of CAL is Sharpe Ratio of risky portfolio \[Sharpe\ Ratio = \frac{E(R_i)-R_f}{\sigma_i}\]
  • Separation Property implies portfolio choice, separated into two tasks
    1. Determine the optimal risky portfolio that contains only risky assets
    2. Determine the allocation to risky portfolio and risk-free asset based on investor’s risk aversion (A)

Conclusion

Conclusion

  • Return and risk refresher
  • Use a utility function to model investor preferences
    • This allows us to combing risk and return into a single equation
    • Allows us to incorporate different levels of risk aversion (A)
  • Portfolio theory
    • What are the risk and return of portfolios?
    • How do investors benefit from diversification?
    • What portfolios maximise investor utility?
      • Without a risk-free asset
      • With a risk-free asset

Portfolio risk and correlation